An elimination lemma for algebras with PBW bases

ABSTRACT

Let K be a field, and $A=K[a_1,\dots ,a_n]$ a finitely generated K-algebra with the PBW K-basis $ℬ=\{a_1^{{\alpha }_1}\cdots a_n^{{\alpha }_n}|({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{N}}^n\}$. It is shown that if L is a nonzero left ideal of A with GK.dim(A∕L) = d<n ( =  the number of generators of A), then L has the elimination property in the sense that V(U)∩L≠{0} for every subset $U=\{a_{i_1},\dots ,a_{i_{d+1}}\}\subset \{a_1,\dots ,a_n\}$ with $i_1<i_2<\cdots <i_{d+1}$, where V(U) = K-span$\{a_{i_1}^{{\alpha }_1}\cdots a_{i_{d+1}}^{{\alpha }_{d+1}}|({\alpha }_1,\dots ,{\alpha }_{d+1})\in {\mathbb{N}}^{d+1}\}$. In terms of the structural properties of A, it is also explored when the condition GK.dim(A∕L)<n may hold for a left ideal L of A. Moreover, from the viewpoint of realizing the elimination property by means of Gröbner bases, it is demonstrated that if A is in the class of binomial skew polynomial rings or in the class of solvable polynomial algebras, then every nonzero left ideal L of A satisfies GK.dim(A∕L)< GK.dimA = n ( =  the number of generators of A), thereby L has the elimination property.

KEYWORDS:

• Elimination
• filtration
• Gelfand-Kirillov dimension
• Gröbner basis
• PBW basis

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 13P10
• Secondary 16W70, 68W30 (16Z05)